⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 060301 Construction of Complete Orthogonal Genuine Multipartite Entanglement State * Feng-Lin Wu (吴风霖)1,2,3, Si-Yuan Liu (刘思远)1,2,3**, Wen-Li Yang (杨文力)1,2, Heng Fan (范桁)1,2,3 Affiliations 1Institute of Modern Physics, Northwest University, Xi'an 710127 2Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127 3Institute of Physics, Chinese Academy of Sciences, Beijing 100190 Received 26 January 2019, online 18 May 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11775177, 11775178, 11647057 and 11705146, the Special Research Funds of the Department of Education of Shaanxi Province under Grant No 16JK1759, the Basic Research Plan of Natural Science in Shaanxi Province under Grant No 2018JQ1014, the Major Basic Research Program of Natural Science of Shaanxi Province under Grant No 2017ZDJC-32, the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province under Grant No 2017KCT-12, the Northwest University Scientific Research Funds under Grant No 15NW26, and the Double First-Class University Construction Project of Northwest University.
**Corresponding author. Email: syliu@iphy.ac.cn Citation Text: Wu F L, Liu S Y, Yang W L and Fan H 2019 Chin. Phys. Lett. 36 060301    Abstract With the development of quantum information processing, multipartite entanglement measures are needed in many cases. However, there are still no complete orthogonal genuine multipartite entanglement (GME) bases available as Bell states to bipartite systems. To achieve this goal, we find a method to construct complete orthogonal GME states, and we exclude many equivalent states by leveraging the group theory. We also provide the case of a $3$-order $3$-dimensional Hilbert space as an example and study the application of general results in the dense coding scheme as an application. Moreover, we discuss some open questions and believe that this work will enlighten extensive studies in this field. DOI:10.1088/0256-307X/36/6/060301 PACS:03.67.Mn, 03.65.Ud © 2019 Chinese Physics Society Article Text Entanglement is an important component in quantum information and a central feature of quantum mechanics.[1,2] Its application holds great potential in quantum information processing in areas such as quantum teleportation,[3,4] quantum cryptography[5] and measurement-based quantum computing.[6] To study bipartite entanglement, a variety of entanglement measures have been defined to quantify entanglement.[7–12] With the development of quantum information processing, bipartite entanglement is no longer suitable in many circumstances. There are various kinds of multipartite entanglement measures that have been proposed,[13–19] and many different multipartite entanglement systems have been investigated.[20–40] However, there is still a problem in that we have no complete orthogonal genuine multipartite entanglement (GME) bases (or states) to measure a multipartite system. Some researchers have provided us with a set of $3$-qubit complete orthogonal states in the study of tripartite teleportation.[41] In Ref.  [3] although the authors obtained the protocol of multipartite teleportation, there was no projector mentioned at all. Given that Bell states are important to bipartite systems and there are no joint-measurement bases available yet in the multipartite system, we aim at finding the general form of joint-measurement bases (or states) in $n$-order $d$-dimensional Hilbert space, as Bell states to a bipartite system. Fortunately, to analyze and quantify the concurrence of arbitrary high-dimensional states, Ma et al. proposed a method called GME-concurrence.[19] For convenience, we need to find a set of maximal GME states to construct multipartite complete orthogonal joint-measurement bases. From the literature,[19] we know that Greenberger–Horne–Zeilinger (GHZ) states are a kind of maximal GME state. However, GHZ states cannot be used to construct complete orthogonal states. Thus we need to construct a set of states, which has the character that after one of the Hilbert subspaces is measured by projector, the remaining part is still a maximal GME state. The results show that these states are ideal bases to compose a series of complete orthogonal GME states. For convenience, we need to find a set of maximal GME states to construct multipartite complete orthogonal joint-measurement bases. From the literature,[19] we know that GHZ states are a kind of maximal genuine multipartite entanglement states. From the definition of GME-concurrence,[19] we can obtain that the maximal GME state must be a pure state. If we have a $d$-dimensional state with $\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1} |i\rangle$, the simplest method to make it entangle with another $d$-dimensional particle would be that each base entangles with the corresponding base. This bipartite state has the form $\frac{1}{\sqrt{d}} \sum_{i=0} ^{d-1} |i i\rangle$. In fact, if we exchange different bases of one subspace arbitrarily, the bipartite state will also be maximally entangled. Thus if we can extend it to greater Hilbert spaces, we will obtain many different GME states. We prepare $n$ sets of initialized orthogonal bases in a $d$-dimensional Hilbert space $\mathcal{H}_n$ (the $d$ index on $\mathcal{H}$ is omitted hereafter). They all have the same form $$\begin{alignat}{1} \!\!\!\!\!&\{|{\it \Psi}\rangle \}_{\mathcal{H}_n}\\ \!\!\!\!\!=\,&\{|i_{r_1}\rangle=|0\rangle, |i_{r_2}\rangle=|1\rangle, \ldots, |i_{r_d}\rangle=|d-1\rangle \}_{\mathcal{H}_1},~~ \tag {1} \end{alignat} $$ where $|i_{r_\alpha}\rangle$ is the $\alpha$th base with increasing order. We will consider a $d$-order permutation group $S_d$ with group elements $$ P_\beta=\left(\begin{matrix} 1 & 2 & 3 & \cdots & d \\ r_{\alpha_1} & r_{\alpha_2} & r_{\alpha_3} & \cdots & r_{\alpha_d} \\ \end{matrix} \right).~~ \tag {2} $$ The group element makes $|i_{r_\alpha}\rangle$ replace $|i_\alpha\rangle$. According to the principle of permutations and combinations, we know that the $d$-order permutation group has $d!$ group elements and $\beta=1,2,3,\ldots,d!$. Then we have $$\begin{align} |\psi_\beta \rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}\!=&\frac{1}{\sqrt{d}}P_\beta^{(\mathcal{H}_2)}\left(\begin{matrix} |i_1\rangle_{\mathcal{H}_2} & |i_2\rangle_{\mathcal{H}_2} & \cdots &|i_d\rangle_{\mathcal{H}_2} \end{matrix} \right) \\ &\cdot\left(\begin{matrix} |i_1\rangle_{\mathcal{H}_1} \\ |i_2\rangle_{\mathcal{H}_1} \\ \vdots \\ |i_d\rangle_{\mathcal{H}_1} \\ \end{matrix}\right).~~ \tag {3} \end{align} $$ It can also be written as $$ |\psi_\beta\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}=\frac{1}{\sqrt{d}} \sum_{\alpha^{(\mathcal{H}_2)}}P_\beta^{(\mathcal{H}_2)}|i_{r_\alpha} \rangle_{\mathcal{H}_2} |i_{r_\alpha}\rangle_{\mathcal{H}_1}.~~ \tag {4} $$ Therefore, we have $d!$ maximal entanglement states in Hilbert space $\mathcal{H}_2 \otimes \mathcal{H}_1$. Then, we can select $d$ maximal entanglement states from $d!$ bases. For two arbitrary bases $|\psi_{\beta_j}\rangle$ and $|\psi_{\beta_{j'}}\rangle$, they must have $\langle \psi_{\beta_j} | \psi_{\beta_{j'}}\rangle=0$. We make the $d$ orthogonal maximal entanglement states be a set as follows: $$\begin{alignat}{1} \!\!\!\!\!&\{|{\it \Psi}_m\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} \} \\ \!\!\!\!\!=\,&\{|\psi^{(m)}_{\beta_1}\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi^{(m)}_{\beta_2}\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}, \ldots, |\psi^{(m)}_{\beta_d}\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} \},~~ \tag {5} \end{alignat} $$ where $m$ indicates that there may be $m$ different combinations. However, because of the duplicate selection for one basis of them, particularly when $d \geq 4$, it is too hard to calculate the maximal value of $m$. With the set of $|{\it \Psi}_m\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}$, we can combine $d!$ sets of maximal entanglement states in Hilbert space $\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1$ as follows: $$\begin{align} &|\psi_\beta\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}\\ =\,&\frac{1}{\sqrt{d}^2} \sum_{\alpha^{(\mathcal{H}_3)}} P_\beta^{(\mathcal{H}_3)} |i_{r_\alpha}\rangle_{\mathcal{H}_3}|\psi^{(m)}_{\beta_{r_\alpha}} \rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}.~~ \tag {6} \end{align} $$ We find $d$ orthogonal states again, and repeat these procedures. For $n$-order $d$-dimensional Hilbert spaces, we have $$\begin{align} |\psi_\beta\rangle_{\otimes_{h=n}^1 \mathcal{H}_h}=\,& \frac{1}{\sqrt{d}^{n-1}}\sum_{\alpha^{(\mathcal{H}_n)}}P_\beta^{(\mathcal{H}_n)}|i_{r_\alpha} \rangle_{\mathcal{H}_n}\\ &\cdot|\psi^{(m)}_{\beta_{r_\alpha}}\rangle_{\otimes_{h=n-1}^{1} \mathcal{H}_h}.~~ \tag {7} \end{align} $$ In this form, we can easily find that $$ \rho_1=\rho_2=\rho_3=\cdots=\rho_n= \frac{\mathbb{I}}{d},~~ \tag {8} $$ where $\rho_h$ is the reduced density matrix at $\mathcal{H}_h$ subspace, and ${\mathbb{I}}$ is the identity matrix. With the symmetry of different Hilbert spaces, no matter which Hilbert space was projected measured, the remaining spaces also have a maximal GME state (see Fig. 1). From the above calculations, we can see that there are only $d$ maximal GME states at a set of $\{{\it \Psi}_m\}$, and every state has $d^{n-1}$ product vectors in a maximal entanglement state with $n$ $d$-dimensional Hilbert space. It is obvious that those orthogonal states can only span a $d$-dimensional space if we have no phase factors. That is to say, they are not complete for $\otimes_{h=n}^1 \mathcal{H}_h$. In this case, some operations like dense coding and teleportation cannot be calculated. Thus it is necessary to introduce phase factors.
cpl-36-6-060301-fig1.png
Fig. 1. Unlike the GHZ states, after project measuring of one subspace, the remaining subspaces of $|\psi_\beta\rangle$ still have maximal GME. (a) If the maximal GME state is a GHZ state, after the project measuring of one subspace, remaining particles will have no entanglement. (b) If the maximal GME state is our constructed state, remaining particles will still be maximal GME states.
cpl-36-6-060301-fig2.png
Fig. 2. The local phase factors. All the phase factors are obtained by local unitary operation. Compared with global phase factors, it will be more convenient for local operation and classical communication (LOCC).
For local phase factors, we need to add a unitary operation for the phase to each subspace $\mathcal{H}_h$, as shown in Fig. 2. From our knowledge of group theory, these local unitary operations are a set of unitary representations of the $d$-order cyclic group $C_d$. The unitary operation has the form $$ U^{(l)}=\sum_{k=0}^{d-1} e^{\frac{-i2kl\pi}{d}}|k\rangle\langle k|,~~ \tag {9} $$ where $k$ and $l$ are integers from 0 to $d-1$, with $l$ implying that the cyclic group $C_d$ has $d$ group elements, and $U^0=\mathbb{I}$. Because the local operation will not change the degree of entanglement, for any $\{|{\it \Psi}_m\rangle_{\otimes_{h=n}^1 \mathcal{H}_n}\}$ we have $$ |\psi_\beta^{(m)}\rangle^{\theta_p}=(\otimes_{h=n}^1 U_h^{(l_{r_h})})|\psi_\beta^{(m)}\rangle,~~ \tag {10} $$ where $h$ is the $h$th Hilbert subspace, $l$ and $m$ are unconstrained, $\theta$ represents the local phase operation, and $p=0,1,2,\ldots, d^n-1$ means that there are $d^n$ different local phase operations. It seems that each Hilbert subspace has $n$ different operations. Yet with the symmetry of $SU(N)$ groups, the states which have the same local phase operation in each subspace are mutually equivalent. This means $$\begin{alignat}{1} &(\otimes_{h=n}^1U_h^{l})|\psi_\beta^{(m)}\rangle^{\theta_p} = (\otimes_{h=n}^1 U_h^{l})(\otimes_{h=n}^1 U_h^{(l_{r_h})})|\psi_\beta^{(m)}\rangle \\ &=(\otimes_{h=n}^1 U_h^{(l_{r_h})})(\otimes_{h=n}^1 U_h^{l})|\psi_\beta^{(m)}\rangle =|\psi_\beta^{(m)}\rangle^{\theta_p}.~~ \tag {11} \end{alignat} $$ We can prove this easily using the commutation of Abel group elements. Therefore, we have $d^n /d$ different $|\psi_\beta^{(m)}\rangle^{\theta_p}$. With the $d$ orthogonal states of $\{|{\it \Psi}_m\rangle\}$, we can obtain a set of complete bases of GME in Hilbert space $\otimes_{h=n}^1 \mathcal{H}_h$. For a state that has the form $\frac{1}{\sqrt{d}^{n-1}}\sum_{r_\alpha=1}^{d^{n-1}} |i_{r_\alpha}\rangle$, we can always denote the phase factor of $|i_{r_1}\rangle$ as $e^{-i 2k\pi}=1$, where $k=1,2,3,\ldots$. For the determined $m$ and $\beta$, we have $$\begin{align} &|\psi_\beta^{(m)}\rangle^{\phi_1}= \frac{1}{\sqrt{d}^{n-1}}( e^{-i2\pi}|\psi_{r_1}\rangle +e^{-i\frac{d^{n-1}-1}{d^{n-1}}2\pi}|\psi_{r_2}\rangle\\ &+e^{-i\frac{d^{n-1}-2}{d^{n-1}}2\pi}|\psi_{r_3}\rangle +\cdots +e^{-i\frac{2\pi}{d^{n-1}}}|\psi_{r_{d^{n-1}}}\rangle ), \\ &|\psi_\beta^{(m)}\rangle^{\phi_2}=\frac{1}{\sqrt{d}^{n-1}}( e^{-i2\cdot2\pi}|\psi_{r_1}\rangle +e^{-i2\frac{d^{n-1}-1}{d^{n-1}}2\pi}|\psi_{r_2}\rangle \\ &+e^{-i2\frac{d^{n-1}-2}{d^{n-1}}2\pi}|\psi_{r_3}\rangle +\cdots +e^{-i\frac{4\pi}{d^{n-1}}}|\psi_{r_{d^{n-1}}}\rangle ), \\ &|\psi_\beta^{(m)}\rangle^{\phi_3}= \frac{1}{\sqrt{d}^{n-1}}( e^{-i3\cdot2\pi}|\psi_{r_1}\rangle +e^{-i3\frac{d^{n-1}-1}{d^{n-1}}2\pi}|\psi_{r_2}\rangle \\ &+e^{-i3\frac{d^{n-1}-2}{d^{n-1}}2\pi}|\psi_{r_3}\rangle +\cdots +e^{-i\frac{6\pi}{d^{n-1}}}|\psi_{r_{d^{n-1}}}\rangle ), \\ &\cdots \\ &|\psi_\beta^{(m)}\rangle^{\phi_{d^{n\!-\!1}}}\!=\!\frac{1}{\sqrt{d}^{n\!-\!1}} (e^{-\!id^{n\!-\!1}2\pi}|\psi_{r_1}\rangle \!+\!e^{-\!i(d^{n\!-\!1}\!-\!1)2\pi}|\psi_{r_2}\rangle\\ &+e^{-i(d^{n-1}-2)2\pi}|\psi_{r_3}\rangle +\cdots +e^{-i2\pi}|\psi_{r_{d^{n-1}}}\rangle ).~~ \tag {12} \end{align} $$ Writing them as a phase unitary operation with $\tilde{U}$, we have $$ |\psi^{(m)}\rangle^{\phi_q}=\tilde{U}^q|\psi\rangle,~~ \tag {13} $$ where $q=0,1,2,\ldots,d^{n-1}-1$, and $\tilde{U}^q$ has the form $$ \tilde{U}^q=\Big(\sum_{k=0}^{d^{n-1}-1} e^{-i\cdot q \cdot 2\pi /d^{n-1}}|(k)_d\rangle\langle (k)_d|\Big) \otimes \mathbb{I}_d,~~ \tag {14} $$ where $(\cdot)_d$ is the $d$-adic representation, and $k$'s are integers. It is obvious that the set of $\tilde{U}^q$ is isomorphic to the $d^{n-1}$-order cyclic group $C_{d^{n-1}}$. The unit element is $\tilde{U}^0=\mathbb{I}_{d^n}$, and the generator is $\tilde{U}^1$. Compared with local phase factors, most global phase factors cannot be constructed by local unitary operation (Fig. 3). Since the $d$-order cyclic group is a proper subgroup of the $d^{n-1}$-order cyclic group, some of the phase factors can be achieved by local operations. It is worth noting that the two kinds of phase factors are equivalent when $n=2$. We will calculate the case of $d=3$ as an example. When $d=3$, the Hilbert space has 3 bases, $|0\rangle$, $|1\rangle$ and $|2\rangle$. Following the first step and adding the Hilbert space $\mathcal{H}_2$ to $\mathcal{H}_1$, we have $$ |\psi_1\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}=\frac{1}{\sqrt{3}} (|00\rangle+|11\rangle+|22\rangle).~~ \tag {15} $$
cpl-36-6-060301-fig3.png
Fig. 3. The global phase factors. Because of the proper subgroup of higher order cyclic groups, some of the phase factors can be achieved by local operations (in comparison of Tables 1 and 2). Most of the global phase factors can only be obtained by global operations.
The $3$-order permutation group $S_3$ has $3!=6$ group elements, and the regular representations as a set of unitary operations could operate the bases of $\mathcal{H}_2$. Those regular representations of permutation group elements are $$\begin{align} D(P_1)=\,&\left(\begin{matrix} 1 & & \\ & 1 & \\ & & 1 \\ \end{matrix} \right), ~ D(P_4)=\left( \begin{matrix} 1 & & \\ & & 1 \\ & 1 & \\ \end{matrix} \right), \\ D(P_2)=\,&\left( \begin{matrix} & 1 & \\ & & 1 \\ 1 & & \\ \end{matrix} \right),~ D(P_5)=\left( \begin{matrix} & & 1 \\ & 1 & \\ 1 & & \\ \end{matrix} \right), \\ D(P_3)=\,&\left( \begin{matrix} & & 1 \\ 1 & & \\ & 1 & \\ \end{matrix} \right), ~ D(P_6)=\left( \begin{matrix} & 1 & \\ 1 & & \\ & & 1 \\ \end{matrix} \right).~~ \tag {16} \end{align} $$ It is well known that local unitary operations do not change the entanglement. Therefore, we have six maximal genuine entanglement states. Using those unitary operations on $\mathcal{H}_2$, we have $$\begin{align} |\psi_1\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} =\,& \frac{1}{\sqrt{3}}(|00\rangle+|11\rangle+|22\rangle), \\ |\psi_2\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} =\,& \frac{1}{\sqrt{3}}(|10\rangle+|21\rangle+|02\rangle), \\ |\psi_3\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} =\,& \frac{1}{\sqrt{3}}(|20\rangle+|01\rangle+|12\rangle), \\ |\psi_4\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} =\,& \frac{1}{\sqrt{3}}(|00\rangle+|21\rangle+|12\rangle), \\ |\psi_5\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} =\,& \frac{1}{\sqrt{3}}(|20\rangle+|11\rangle+|02\rangle), \\ |\psi_6\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}=\,& \frac{1}{\sqrt{3}}(|10\rangle+|01\rangle+|22\rangle).~~ \tag {17} \end{align} $$ Then, we select three bases to be a set of $\{|{\it \Psi}_m\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}\}$. We can easily obtain that the utmost value of $m$ is 2, and $$\begin{align} \{|{\it \Psi}_1\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}\} =\,&\{ |\psi_1\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_2\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_3\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} \}, \\ \{|{\it \Psi}_2\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}\} =\,&\{ |\psi_4\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_5\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_6\rangle_{\mathcal{H}_2\otimes\mathcal{H}_1} \}.~~ \tag {18} \end{align} $$ Considering the phase factors, we have two sets of complete orthogonal bases, which have 18 maximal genuine entanglement states. Removing the phase factors, adding the Hilbert space $\mathcal{H}_3$ and repeating the previous step, there will be $$\begin{align} &|\psi_1\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|000\rangle+|011\rangle+|022\rangle+|110\rangle\\ &
+|121\rangle+|102\rangle +|220\rangle+|201\rangle+|212\rangle), \\ &|\psi_2\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|100\rangle+|111\rangle+|122\rangle+|210\rangle\\ &
+|221\rangle+|202\rangle +|020\rangle+|001\rangle+|012\rangle), \\ &|\psi_3\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|200\rangle+|211\rangle+|222\rangle+|010\rangle\\ &
+|021\rangle+|002\rangle +|120\rangle+|101\rangle+|112\rangle), \\ &|\psi_4\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|200\rangle+|211\rangle+|222\rangle+|110\rangle\\ &
+|121\rangle+|102\rangle +|020\rangle+|001\rangle+|012\rangle), \\ &|\psi_5\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|100\rangle+|111\rangle+|122\rangle +|010\rangle\\ &
+|021\rangle+|002\rangle +|220\rangle+|201\rangle+|212\rangle), \\ &|\psi_6\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|000\rangle+|011\rangle+|022\rangle +|210\rangle\\ &
+|221\rangle+|202\rangle +|120\rangle+|101\rangle+|112\rangle), \\ &|\psi_7\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|020\rangle+|011\rangle+|002\rangle +|110\rangle\\ &
+|101\rangle+|122\rangle +|200\rangle+|221\rangle+|212\rangle), \\ &|\psi_8\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|120\rangle+|111\rangle+|102\rangle +|210\rangle\\ &
+|201\rangle+|222\rangle +|000\rangle+|021\rangle+|012\rangle), \\ &|\psi_9\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|220\rangle+|211\rangle+|202\rangle +|010\rangle\\ &
+|001\rangle+|022\rangle +|100\rangle+|121\rangle+|112\rangle), \\ &|\psi_{10}\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|220\rangle+|211\rangle+|202\rangle +|110\rangle\\ &
+|101\rangle+|122\rangle +|000\rangle+|021\rangle+|012\rangle), \\ &|\psi_{11}\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|120\rangle+|111\rangle+|102\rangle +|010\rangle\\ &
+|001\rangle+|022\rangle +|200\rangle+|221\rangle+|212\rangle), \\ &|\psi_{12}\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} = \frac{1}{3}(|020\rangle+|011\rangle+|002\rangle +|210\rangle\\ &
+|201\rangle+|222\rangle +|100\rangle+|121\rangle+|112\rangle).~~ \tag {19} \end{align} $$ We have $$\begin{align} &\{|{\it \Psi}_1\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}\}\\ =\,&\{|\psi_1\rangle_{\mathcal{H}\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_2\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_3\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} \}, \\ &\{|{\it \Psi}_2\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}\}\\ =\,&\{|\psi_4\rangle_{\mathcal{H}\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_5\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_6\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} \}, \\ &\{|{\it \Psi}_3\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}\}\\ =\,&\{|\psi_7\rangle_{\mathcal{H}\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_8\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_9\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} \}, \\ &\{|{\it \Psi}_4\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}\}\\ =\,&\{|\psi_{10}\rangle_{\mathcal{H}\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_{11}\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1}, |\psi_{12}\rangle_{\mathcal{H}_3\otimes\mathcal{H}_2\otimes\mathcal{H}_1} \}.~~ \tag {20} \end{align} $$ Adding a new subspace and phase factors, we can obtain the complete orthogonal GME states of Hilbert space $\otimes_{h=n}^1 \mathcal{H}_h$. Taking $|\psi_1\rangle$ as an example, the results are listed in Tables 1 and 2. In addition, for $d=4$, we choose two $\{|{\it \Psi}_m\rangle\}$ as follows: $$\begin{align} &\{|{\it \Psi}_1\rangle\}\\ =\,& \left\{\begin{matrix} |\psi_1^1\rangle=\frac{1}{2}(|000\rangle+|011\rangle+|022\rangle+|033\rangle) \\ |\psi_2^1\rangle=\frac{1}{2}(|110\rangle+|101\rangle+|132\rangle+|123\rangle) \\ |\psi_3^1\rangle=\frac{1}{2}(|220\rangle+|231\rangle+|202\rangle+|213\rangle) \\ |\psi_4^1\rangle=\frac{1}{2}(|330\rangle+|321\rangle+|312\rangle+|313\rangle) \end{matrix}\right\},~~ \tag {21} \end{align} $$ $$\begin{align} &\{|{\it \Psi}_2\rangle\}\\ =\,&\left\{\begin{matrix} |\psi_1^2\rangle=\frac{1}{2}(|000\rangle+|011\rangle+|022\rangle+|033\rangle) \\ |\psi_5^2\rangle=\frac{1}{2}(|110\rangle+|121\rangle+|132\rangle+|103\rangle) \\ |\psi_3^2\rangle=\frac{1}{2}(|220\rangle+|231\rangle+|202\rangle+|213\rangle) \\ |\psi_6^2\rangle=\frac{1}{2}(|330\rangle+|301\rangle+|312\rangle+|333\rangle) \\ \end{matrix} \right\}.~~ \tag {22} \end{align} $$ Although $\{|{\it \Psi}_1\rangle\}$ and $\{|{\it \Psi}_2\rangle\}$ are inequivalent, $|\psi_1\rangle$ and $|\psi_3\rangle$ appear in two different sets at the same time. That is the reason why the number of $\{|{\it \Psi}_m\rangle \}$ is hard to calculate when $d\geq 4$. Another application is dense coding. A dense coding scheme is that Alice and Bob share a pair of entangled qubits. Alice rotates her qubit by four different operations $O=UP$, and then sends her qubit to Bob. Bob will perform joint measurement on these two qubits when he receives the qubit from Alice. In this case, Alice can send $\log_2 4=2$ bits of information to Bob each time. Based on this case, we find that GME states will be more efficient. Consider the case that Alice and Bob share $n$ particles in $d$-dimensional maximal GME states of $\{|{\it \Psi}_m\rangle\}_{\otimes_{h=n}^{1}\mathcal{H}_h}$ with local phase factors. Alice has $a$ particles and Bob has $b$ particles, where $a+b=n$. Alice operates her particles by operations $O=UP$ in each subspace, and then gives Bob her particles. Bob will measure those $n$ particles by joint measurement. Thus Bob could know the operations Alice has carried out. Obviously, Alice has $(d\cdot d!)^a$ different operators $O$, but there are many equivalent operations actually. Because of the unknown number of $m$, we only consider the case of ensured $\{|{\it \Psi}_m\rangle\}$. In this constraint, for $d$ orthogonal states in the set of $\{|{\it \Psi}_m\rangle\}$, there are $d$ different kinds of $P$. Then Alice could only transfer $\log_2 d$ bits of information to Bob by operator $P$. The operator $U$ for any particles will be $d$ different operations on Alice's particles. That is to say, there are $\log_2 d^a$ bits of information that will be transferred to Bob by operator $U$. This means that Alice can use her $a$ particles to transfer $\log_2 (d \cdot d^a) = \log_2 d^{a+1} = (a+1)\log_2 d$ bits of information to Bob in the dense coding case. The maximum efficiency is reached when $b=1$, and Alice can transfer $\log_2 d^n$ bits of information to Bob. In the case of two-particle dense coding, although the particles Alice sent to Bob are invariable, Bob must measure $a$ times with $a$ pairs of particles, and each time can only obtain $\log_2 d$ bits of information. With maximally complete GME states, Bob needs less prior particles and only one joint measurement.
Table 1. The local phase factors for $|\psi_1\rangle$, where $\omega=e^{-i\frac{2\pi}{3}}$ and $U^1 \otimes U^0 \otimes U^0=U^2 \otimes U^1 \otimes U^1=U^0 \otimes U^2 \otimes U^2$, etc.
$\theta_p$ $|000\rangle$ $|011\rangle$ $|022\rangle$ $|110\rangle$ $|121\rangle$ $|102\rangle$ $|220\rangle$ $|201\rangle$ $|212\rangle$
$U^0 \otimes U^0 \otimes U^0$ 1 1 1 1 1 1 1 1 1
$U^0 \otimes U^0 \otimes U^1$ 1 $\omega$ $\omega^2$ 1 $\omega$ $\omega^2$ 1 $\omega$ $\omega^2$
$U^0 \otimes U^0 \otimes U^2$ 1 $\omega^2$ $\omega$ 1 $\omega^2$ $\omega$ 1 $\omega^2$ $\omega$
$U^0 \otimes U^1 \otimes U^0$ 1 $\omega$ $\omega^2$ $\omega$ $\omega^2$ 1 $\omega^2$ 1 $\omega$
$U^0 \otimes U^1 \otimes U^1$ 1 $\omega^2$ $\omega$ $\omega$ 1 $\omega^2$ $\omega^2$ $\omega$ 1
$U^0 \otimes U^1 \otimes U^2$ 1 1 1 $\omega$ $\omega$ $\omega$ $\omega^2$ $\omega^2$ $\omega^2$
$U^0 \otimes U^2 \otimes U^0$ 1 $\omega^2$ $\omega$ $\omega^2$ $\omega$ 1 $\omega$ 1 $\omega^2$
$U^0 \otimes U^2 \otimes U^1$ 1 1 1 $\omega^2$ $\omega^2$ $\omega^2$ $\omega$ $\omega$ $\omega$
$U^0 \otimes U^2 \otimes U^2$ 1 $\omega$ $\omega^2$ $\omega^2$ 1 $\omega$ $\omega$ $\omega^2$ 1
Table 2. The global phase factors for $|\psi_1\rangle$, where $\lambda=e^{-i\frac{2\pi}{9}}$. Notably, since the $3$-order cyclic group is a proper subgroup of the $9$-order cyclic group, there are three operations $\tilde{U}^0$, $\tilde{U}^3$ and $\tilde{U}^6$ that can be achieved by local phase operations.
$\phi_q$ $|000\rangle$ $|011\rangle$ $|022\rangle$ $|110\rangle$ $|121\rangle$ $|102\rangle$ $|220\rangle$ $|201\rangle$ $|212\rangle$
$\tilde{U}^0$ 1 1 1 1 1 1 1 1 1
$\tilde{U}^1$ 1 $\lambda$ $\lambda^2$ $\lambda^3$ $\lambda^4$ $\lambda^5$ $\lambda^6$ $\lambda^7$ $\lambda^8$
$\tilde{U}^2$ 1 $\lambda^2$ $\lambda^4$ $\lambda^6$ $\lambda^8$ $\lambda$ $\lambda^3$ $\lambda^5$ $\lambda^7$
$\tilde{U}^3$ 1 $\lambda^3$ $\lambda^6$ 1 $\lambda^3$ $\lambda^6$ 1 $\lambda^3$ $\lambda^6$
$\tilde{U}^4$ 1 $\lambda^4$ $\lambda^8$ $\lambda^3$ $\lambda^7$ $\lambda^2$ $\lambda^6$ $\lambda$ $\lambda^5$
$\tilde{U}^5$ 1 $\lambda^5$ $\lambda$ $\lambda^6$ $\lambda^2$ $\lambda^7$ $\lambda^3$ $\lambda^8$ $\lambda^4$
$\tilde{U}^6$ 1 $\lambda^6$ $\lambda^3$ 1 $\lambda^6$ $\lambda^3$ 1 $\lambda^6$ $\lambda^3$
$\tilde{U}^7$ 1 $\lambda^7$ $\lambda^5$ $\lambda^3$ $\lambda$ $\lambda^8$ $\lambda^6$ $\lambda^4$ $\lambda^2$
$\tilde{U}^8$ 1 $\lambda^8$ $\lambda^7$ $\lambda^6$ $\lambda^5$ $\lambda^4$ $\lambda^3$ $\lambda^2$ $\lambda$
In summary, we have investigated the complete orthogonal GME states to construct sets of joint-measurement bases. They are all maximal GME states. Firstly, we obtain the form without phase factors using the $d$-order permutation group $S_d$. Obviously, it is not complete. We find two kinds of phase factors and both of them can make these bases complete. We then calculate the case of $d=3$ and $n=3$ as an example, and consider dense coding as an application. There are some interesting open problems. In the past few years, many kinds of multipartite or multidimensional teleportation have been discussed.[42–45] We speculate that the complete orthogonal GME states will have some extended applications. Moreover, local operation and classical communication (LOCC) plays a fundamental role in entanglement theory. Some researchers have found that the stochastic LOCC classification of pure states can be used to obtain some classification of mixed states.[46,47] Since the orthogonal complete GME states with local phase factors naturally correspond to LOCC, we think that it could be useful in this field. In addition, in view of the importance of Bell states, we believe that this work will enlighten extensive studies, such as the Zeno effect, entanglement distribution and quantum steering. We will discuss those issues in future works. We acknowledge Pei Wang for the helpful suggestion. References Quantum entanglementEntanglement detectionTeleportation and Dense Coding with Genuine Multipartite EntanglementTeleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channelsQuantum cryptography based on Bell’s theoremA One-Way Quantum ComputerMixed-state entanglement and quantum error correctionEntanglement measures and purification proceduresEntanglement of a Pair of Quantum BitsUniversal state inversion and concurrence in arbitrary dimensionsVolume of the set of separable statesComputable measure of entanglementDistributed entanglementConstructing N -qubit entanglement monotones from antilinear operatorsExperimental determination of entanglement for arbitrary pure statesGeometric measure for entanglement in N -qudit pure statesComputation of the geometric measure of entanglement for pure multiqubit statesEntanglement criterion for multipartite systems based on quantum Fisher informationMeasure of genuine multipartite entanglement with computable lower boundsDynamics of multipartite quantum correlations in the qubit-reservoir systemStructure of Multidimensional Entanglement in Multipartite SystemsConstructing genuinely entangled multipartite states with applications to local hidden variables and local hidden states modelsTopological and nematic ordered phases in many-body cluster-Ising modelsEntanglement robustness against particle loss in multiqubit systemsMultipartite entanglement in spin chains and the hyperdeterminantSemi-device-independent characterization of multipartite entanglement of states and measurementsGeometric measures of entanglement in multipartite pure states via complex-valued neural networksMultipartite Entanglement at Finite TemperatureEfficient entanglement purification protocols for d -level systemsGenuine multipartite concurrence for entanglement of Dirac fields in noninertial framesDetecting high-dimensional multipartite entanglement via some classes of measurementsEffect of atmospheric turbulence on entangled orbital angular momentum three-qubit stateFour-partite Bell inequalities based on quantum coherenceConnecting Quantum Contextuality and Genuine Multipartite Nonlocality with the Quantumness WitnessQuantum State Transfer among Three Ring-Connected AtomsDistillability Sudden Birth of Entanglement for Qutrit-Qutrit SystemsMajorization Relation in Quantum Critical SystemsSeparability Criterion for Bipartite States and Its Generalization to Multipartite SystemsMultipartite entanglement for entanglement teleportationProbabilistic teleportation of an unknown entangled state of two three-level particles using a partially entangled state of three three-level particlesProbabilistic Teleportation of an Arbitrary Three-Level Two-Particle State and Classical Communication CostTeleportation of the three-level three-particle entangled state and classical communication costTeleportation of Three-Level Multi-partite Entangled State by a Partial Three-Level Bipartite Entangled StateClassification of Mixed Three-Qubit StatesMultipartite entanglement in 2 × 2 × n quantum systems
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